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User blog:Naruyoko/My Tottaly Not Broken "Ordinal Notation", Which Totally is Well-Defined and Covers All Ordinals To the Limit Ok Why Is the Title so Long
Ok. I'm bad at this. Please a9kqw{#=Mz0:1. \(\lt\) is partial well-ordering relation of two expressions. 0:2. For any expressions \(a\), \(b\), and \(c\), \(a\lt b\land b\lt c\implies a\lt c\). 0:3. For any expressions \(a\) and \(b\), \(a=b\) means that it is equivalent, and it can be interchanged. Or, \(a=b\implies\forall P[P(a)\iff P(b)\) 0:4. For any expression \(a\), \(a=a\). 0:5. For any expressions \(a\) and \(b\), \(a\lt b\oplus b\lt a\oplus a=b\). P1 1:1. The expression consists of \(t\), which then must be followed by \((\), which must then be followed by a \()\), or another expression of this, then \()\). For example, \(t(t())\). 1:2. \(r\), if \(r\ne t()\) then \(t()\lt r\). 1:3. \(r\lt t®\), where \(r\) is an expression. The strength of this is likely \(\omega\)(\(t()\lt t(t())\lt t(t(t()))\lt\cdots\)). P2 2:1. The expression consists of \(t\), which can have superscript of an expression \(\lt t(0,1)\), then must be followed by \((\), then an array, then \()\). If there is an superscript for \(t\), then there must be a \(*\) in expression. For example, \(t(t(),t(t()^{t(t(t()))}))\). 2:2. An array is either nothing, an expression, \(*\), expression with superscript of an expression \(\lt t(0,1)\), a variable name with superscript of an expression \(\lt t(0,1)\), \(*\) with superscript of an expression \(\lt t(0,1)\), or an array followed by a \(,\) and then an array. 2:3. For variable+superscript array \(a^i\), \(a_{t(t())}\) represents the leftmost expression, and \(a_{t®}\) for expression \(r\) is the leftmost experssion right of \(a_r\), seperated by \(,\) when \(a^i\) is fully expanded into form without superscripts, for subscript \(n\) which \(t()\lt n\lt t(i)\). 2:4. \(t(a^n,0)=a^n\) 2:5. \(t(a^n,)=a^n\) 2:6.1:2. \(r\), if \(r\ne t()\) then \(t()\lt r\). 2:7.1:3. \(r\lt t®\), where \(r\) is an expression. 2:8. \(t(t(a),r)=t(t(a,r))\), where \(a\) is an expression and \(r\) is an array. 2:9. \(a^{t(t())}\) is interchangable to \(a\). 2:10. \(a^{t()}\) is interchangeable to nothing. 2:11. \(a^{t®}\) where r is an expression is interchangeable to \(a,a^r\). 2:12. \(t^{t®}(s)\), where \(s\) is an array will be solved as following: 2:12.1. Find all \(*\) in \(s\). 2:12.2. Replace all \(*\) in \(s\) with \(t^r(s)\). Call new array \(s'\). 2:12.3. New expression equivalent to the original is \(t(s')\). 2:13. \(t^{t()}®=t()\) where \(r\) is an expression. 2:14. \(\forall n(n\lt t(0,1)\implies t^n(t()^r,*,a,b^c)\lt t(t()^{t®},a,b^c)\). 2:15.Overrides 2:14. \((t()\lt t(0,1))\land\forall no\implies t(n)\lt t(0,1)\). 2:16. For expressions \(o=t(a^i)\) and \(p=t(b^i)\)... 2:16.1. Let \(k=i\). 2:16.2. If \(a_k\lt b_k\), \(o\lt p\). 2:16.3. If \(b_k\lt a_k\), \(p\lt o\). 2:16.4. If \(k=t()\), \(o=p\). 2:16.5. Replace \(k\) withsuch that \(t(k)=m\) where \(m\) is \(k\) defined in the last iteration of 2:16.1. or 2:16.5. 2:16.6. Go to 2:16.2.. The strength is... I don't know, I think it's like \(\omega^\omega\) or something, considering it kinda mimics Array Notation.\(\small{\(t()\lt t(t())\lt t(t(t()))\lt\\\cdots\lt t(t(),t(t()))\lt t(t(t())^{t(t())})\lt t(t(t(t())),t(t()))\lt\\\cdots\lt t^{t(t(t()))}(*,t(t()))\lt t(t(t(t())^{t(t(t()))}),t(t()))\lt\\\cdots\lt t^{t(t(t(t())))}(*,t(t()))\lt\\\cdots\lt t^{t(t(t(t(t()))))}(*,t(t()))\lt\\\cdots\lt t(t(),t(t(t())))\lt t(t(t()),t(t(t())))\lt\\\cdots\lt t(t(t(),t(t())),t(t(t())))\lt\\\cdots\lt t(t^{t(t(t()))}(*,t(t())),t(t(t())))\lt\\\cdots\lt t^{t(t(t()))}(*,t(t(t())))\lt\\\cdots\lt t(t(),t(t(t(t()))))\lt\\\cdots\lt t(t()^{t(t(t()))},t(t()))\lt t(t(t()),t(),t(t()))\lt\\\cdots\lt t(t(t(),t(t())),t(),t(t()))\lt\\\cdots\lt t^{t(t(t()))}(*,t(),t(t()))\lt\\\cdots\lt t(t(),t(t())^{t(t(t()))})\lt\\\cdots\lt t^{t(t(t()))}(t(),*,t(t()))\lt\\\cdots\lt t(t()^{t(t(t()))},t(t(t())))\lt\\\cdots\lt t(t()^{t(t(t()))},t(t(),t(t())))\lt\\\cdots\lt t^{t(t(t()))}(t()^{t(t(t()))},*)\lt\\\cdots\lt t(t()^{t(t(t(t())))},t(t()))\lt\\\cdots\lt t(t()^{t(t(t(t(t()))))},t(t()))\lt\cdots}\) P3 Oh wow this is tttttterrible. Also I should get myself to be formal. Category:Blog posts